Calculus For Everyone
The Fundamentals Of Limit
A Concept That Changed the World
What Is Limit?
Basically, the meaning of limit in mathematics is the same as the meaning of limit in the real world. Limit means boundary. What does it mean?
Just as a reminder, in the function f(x)=2x+3, x is called the domain and 2x+3 is called the codomain. Now, in the concept of limit, we limit the value of x (domain) to approach a value, let’s call it the value of "a". Why just approach? Why don’t we just write x=a? This is because in some math and physics problems, sometimes we cannot substitute the value of “a” into the variable x. For some reason, sometimes we can only calculate the value of a function whose x value is close to the value of “a”. What is an example? Try to solve the following questions!
If you calculate it correctly, you will find the result y = 0/0 , which means the result is undefined. True, the result is undefined. Well, this is where the concept of limits works. We can’t calculate the value of y when x = 1, but we can calculate the value of y when x approaches 1. The word "approach" here means that the value of x is very close to 1. It can be interpreted that the value of x approaching 1 is equal to 0.9999999999… or 1.000000000000001. Both values are so close to 1 that they represent the value of y when x is equal to 1.
In the graph above, we can still see the value of y at x=1 because that value is the limit value, namely the limit of y when x approaches 1. The value of y=2 is not the exact value when x=1. When x=1, the value of y is still undefined, but we can use the limit value.
We cannot say that the value of y =2 when x=1 because actually, the value of y is undefined when x=1. However, we know that when the value of x is very close to 1, then the value of y is very close to 2. To express this, mathematicians use the concept of limit. So, the above statement can be expressed as:
The limit value of y as x approaches 1 is equal to 2
We can write that statement as follows :
Lim stands for limit. x -> 1 means "the value of x is close to 1". Here is the general form of the limit concept:
The above equation means "the limit value of f(x) as x approaches the value of a is k". k here can be a constant or a polynomial.
There is a condition that the limit f(x) when x=a is equal to k, that is, the left limit and the right limit must be the same.
The left-hand limit is the value of y as x approaches a from the left. For example a=1, then the left limit is the value of y [f(x)] when x approaches 1 from the left, i.e. x=0.9999999999….
The right-hand limit is the value of y as x approaches a from the right. In the example above, the right limit is the value of y when x approaches 1 from the right, i.e. y=2.000000…..1, While the left limit is the value of y when x approaches 1 from the left, i.e. y=1.9999999…
Maybe it will be easier to understand if we look at the graph directly. Take a look at the following graph:
Why is Limit Important?
The concept of limit is used in derivatives and integrals that utilize the concept of "small parts". Okay, maybe you don’t understand, that’s why I’ll give you another example. Have you ever read my post “can you solve this riddle?— Zeno’s Paradox” ?. In that article, I indirectly use the concept of limit, which is the limit for x approaching infinity
How to Solve Limit Problems?
- Substitution. For the limit of f(x) for x close to a, we can calculate the value of the limit by directly substituting x=a so that the limit value is f(a).
- There are times when substitution cannot be done. For example, in the limit problem (x²-1)/(x-1) for x is close to 1. If we substitute directly, the result is 0/0. for that, we need to perform certain operations that theoretically do not change the value of this function. At the limit (x²-1)/(x-1), we can translate it into (x-1)(x+1)/(x-1), so we get the result x+1. Now, after that, we can substitute the value of x=1 in x+1, and the result is 2. So, the limit of the function f(x)=(x²-1)/(x-1) for x close to 1 is equal to 2.
Limit Value of General Function
The general functions referred here are functions such as f(x)=sin (x), f(x) = cos (x), and so on. The value of the limit of this function is important for us to know because it will be widely applied to mathematical problems later.
1. Limit Of (x) = sin (x) / x
2. Limit Of f (x) = cos (x)
3. Limit Of f (x) = tan (x) / x
I will not explain the process of finding the limit value above in this article, because it is a long process. You can find steps to find the limit value on the internet.
Exercise
Can you solve this problem?
Thank you for reading this article. If you have any questions, don’t hesitate to ask in the response section.
In the next article, I will start a discussion about derivatives. See you!
